Mobility of Polymer-Tethered Nanoparticles in Unentangled Polymer Melts

A scaling theory is developed for the motion of a polymer-tethered nanoparticle (NP) in an unentangled polymer melt. We identify two types of scaling regimes depending on the NP diameter d and the size of a grafted polymer chain (tail) R . In one type of regimes, the tethered NP motion is dominated by the bare NP, as the friction coefficient of the tails is lower than that of the less mobile particle. The time dependence of the mean square displacement (MSD) of the tethered NP ⟨Δr ²(t)⟩ in the particle-dominated regime can be approximated by ⟨Δr ²(t)⟩ for the bare NP. In the other type of regimes, the tethered NP motion is dominated by the tails when the friction coefficient of the tails surpasses that of the particle at times longer than the crossover time τ ^∗. In a tail-dominated regime, the MSD ⟨Δr ²(t)⟩ ≈ ⟨Δr ²(t)⟩ only for t < τ ^∗. ⟨Δr ²(t)⟩ of a single-tail NP for t > τ ^∗ is approximated as the MSD ⟨Δr ²(t)⟩ of monomers in a free tail, whereas ⟨Δr ²(t)⟩ of a multi-tail NP for t > τ ^∗ is approximated as the MSD ⟨Δr ²(t)⟩ of the branch point of a star polymer. The time dependence of ⟨Δr ²(t)⟩ in a tail-dominated regime exhibits two qualitatively different sub-diffusive regimes. The first sub-diffusive regime for t < τ ^∗ arises from the dynamical coupling between the particle and the melt chains. The second sub-diffusive regime for t > τ ^∗ occurs as the particle participates in the dynamics of the tails. For NPs with loosely grafted chains, there is a Gaussian brush region surrounding the NP, where the chain strands in Gaussian conformations undergo Rouse dynamics with no hydrodynamic coupling. The crossover time τ ^∗ for loosely grafted multi-tail NPs in a tail-dominated regime decreases as the number of tails increases. For NPs with densely grafted chains, the tails are hydrodynamically coupled to each other. The hydrodynamic radii for the diffusion of densely grafted multi-tail NPs are approximated by the sum of the particle and tail sizes.